being an increasing function.
For \(\pi ,\eta >0\) , each function (REF ) is also strictly increasing and the KR rearrangement becomes an invertible map.
The existence of the KR rearrangement stem from
intuitive factorization properties of any density, e.g., if \(Z=(Z_1,\ldots ,Z_n)\sim \pi \) , then
\(\pi = \prod _k \pi _{Z_k \vert Z_{1:k-1}}\) .
In fact, perhaps the most important property
for our analysis is that
the rearrangement
provides an implicit
characterization of
the marginal conditionals of \(\pi \) , whenever
\(\eta = \mathcal {N}({\bf 0}, {\bf I}_n)\) .
In this case, one can show that each one dimensional mapping
(REF )
pushes forward
the marginal conditional
\(\xi \mapsto \pi _{Z_k\vert Z_{1:k-1}}(\xi \vert z_{1:k-1})\)  to a one dimensional standard normal,
for all \(z_{1:k-1}\in \mathbb {R}^{k-1}\)  [1]}.
We leverage this property in the next section, where we discuss our first filtering algorithm, while we
defer questions pertaining to the estimation of
the rearrangement to Section .
